Queueing system MAP|PH|N|N+R with impatient heterogeneous customers as a model of call center

A multi-server queueing system with a Markovian arrival process (MAP), a finite buffer and impatient heterogeneous customers useful in modeling a call center is investigated. The servers are identical and independent of each other. The service time of a customer has a Phase type distribution (PH). If all servers are busy at a customer arrival epoch, the customer moves to the buffer with probability, which depends on the number of customers in the system, or, alternatively, leaves the system forever. During a waiting period the customers can be impatient and leave the system. An efficient algorithm for calculating the stationary probabilities of system states is proposed. A special approach for reducing the system dimension is used. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distribution are derived. Some illustrative numerical results are presented.